Transmit Power Allocation for BER Performance Improvement in Multicarrier Systems

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1 Transmit Power Allocation for Performance Improvement in Systems Chang Soon Par O and wang Bo (Ed) Lee School of Electrical Engineering and Computer Science, Seoul National University parcs@mobile.snu.ac.r, lee@snu.ac.r Abstract In a wireless multicarrier system, transmit power allocation over different subchannels is an effective means to improve the performance. In this paper, the optimal transmit power allocation scheme is developed to improve the bit error rate () performance in a multicarrier system with diversity reception. A simple suboptimal scheme is also derived from the optimal one, and an asymptotic case is discussed. Numerical results show that the optimal and suboptimal power allocation schemes significantly outperform the equal power allocation scheme. The effects of the modulation level, the number of receiving antennas, and the number of subchannels on the performance are also investigated. I. INTRODUCTION communication system is promising for future wideband wireless communications, and recently the system is being applied to several fixed and mobile radio systems, such as digital audio and video broadcasting, and wireless LAN [], []. In a multicarrier system, a wideband channel is divided into multiple narrowband subchannels using orthogonal subcarriers, and multiple data streams are transmitted in parallel through these subchannels. In a multicarrier system, it is natural to allocate equal transmit power to multiple subchannels, when the channel state information is not available at the transmitter. When the channel state information is available at the transmitter, however, effective transmit power allocation may improve error rate performance or increase the capacity. Transmit power allocation based on the water-filling solution has been studied in [3] and [4], and it has been combined with adaptive modulation to maximize the capacity [5], [6]. These schemes of increasing capacity may be suitable for variable rate services such as and web browsing. On the contrary, delay-sensitive services such as voice or video are usually provided at a fixed rate. In these applicases, it is desirable to design a transmit power allocation scheme that improves error rate performance for a given rate. In this paper, we develop the optimal transmit power allocation scheme to improve bit error rate () performance in a multicarrier system with receive antenna diversity. Receive antenna diversity is used to mitigate the effects of fading [], []. Based on the optimal scheme, a computationally efficient suboptimal scheme is also derived. Furthermore, it is shown that an asymptotic case of the suboptimal scheme corresponds to the equal signalto-noise ratio (SNR) scheme, by which the received SNR becomes the same for all subchannels. The performance of the proposed power allocation schemes is evaluated, and compared with that of the equal power allocation scheme. The effects of the modulation level, the number of receiving antennas, and the number of subchannels on the performance are also investigated. Numerical results show that the use of the optimal or suboptimal power allocation scheme significantly improves the performance of a multicarrier system. The equal SNR scheme is found to perform well in the presence of large number of receiving antennas. The remainder of this paper is organized as follows. Section II describes the system and channel models. In Section III, the optimal transmit power allocation scheme is derived, and a simple suboptimal scheme is derived from the optimal one. An asymptotic case of the suboptimal scheme is also investigated in Section III. Numerical results are presented in Section IV, and conclusions are drawn in Section V. II. SYSTEM AND CHANNEL MODELS A wireless multicarrier system considered in this paper is depicted in Fig.. An input data stream is divided into parallel substreams through a serial-to-parallel converter. The transmit power p is assigned to the th substream ( =,,, ). In the multicarrier modulator, the substreams are modulated on orthogonal subcarriers to form a transmit signal. The receiver is equipped with N antennas, from which N replicas of the transmit signal for each subcarrier are received. It is assumed that the signals received from different antennas experience independent, slowly varying, frequency-nonselective Rayleigh fading. The output of a multicarrier for the th subcarrier at the nth receive antenna may be expressed as y, n = p h, nd + n, n, =,,,, n =,,, N () where h,n denotes the multiplicative fading coefficient for

2 the th subcarrier at the nth antenna, and they are assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. n,n s represent the additive white Gaussian noise (AWGN), and they are assumed to be i.i.d. complex Gaussian random variables with zero mean and variance of. d is the encoded data symbol with unit average power, and p is the transmit power for the th subcarrier with the total power constraint given as p = P, p () = where P denotes the average transmit power per subcarrier, when the total transmit power is equally distributed into the subcarriers. As shown in Fig., signals received from N antennas are combined for each subcarrier to achieve antenna gain and diversity gain. It is assumed that the maximal ratio combining () is employed to maximize the SNR []. The transmitted symbol on each subcarrier is estimated based on the output of each. After, the SNR for the th subcarrier may be calculated as γ = α p (3) where N n =, n α h σ is the ratio of the combined channel gain for the th subchannel to the noise power, representing the overall channel state for the th subchannel. The channel states ( =,,, ) are required to determine the transmit power p ( =,,, ), and they are assumed to be perfectly nown to the transmitter. These channel states can be obtained by feedbac from the receiver in a frequency division duplex (FDD) system, or can be estimated at the transmitter in a time division duplex (TDD) system. III. TRANSMIT POWER ALLOCATION In this section, several transmit power allocation schemes are described. The equal power allocation scheme is briefly discussed in Section III-A, and power allocation scheme that is optimal in terms of the is developed in Section III-B. A suboptimal scheme is derived as a simplified version of the optimal scheme in Section III-C, and the equal SNR scheme is discussed as an asymptotic case of the suboptimal scheme in Section III-D. A. Equal Power Allocation When the channel state information is not available at the transmitter, it is natural to allocate equal transmit power to subcarriers. This scheme is referred to as the equal power allocation scheme. In this scheme, the transmit power for each subcarrier is P in (), and the SNR γ for the th subcarrier is α P. When the channel state information is available at the transmitter, the total transmit power may be allocated in a more effective way to achieve better performance, as described in the following subsections. B. Power Allocation To derive the optimal power allocation scheme, we first express the overall as a function of the transmit power of subcarriers, { p =,,, }, and then find { p } that minimizes the overall. The for the th subcarrier is generally a function of the SNR γ, and thus the Pb ( e α ) for a given channel state α may be expressed as P ( e α ) = f α p, =,,, (4) b where f ( ) is a function determined by a specific modulation scheme. Since data streams are transmitted over independent subchannels, the overall for given channel states of { α } can be calculated as an arithmetic mean of P ( e α ) in (4): b P ( e α, α,, α ) = f α p. (5) b = Note that the average becomes minimal when the in (5) is minimized for each given channel state. To find the optimal {p } that minimizes (5), we use the Lagrange multiplier method with the total power constraint in (). The Lagrangian function may be expressed as J ( p, p,, p ) = f ( α p ) + λ p P (6) = = where λ denotes the Lagrange multiplier. By differentiating (6) with respect to p and setting it to zero, we obtain a set of equations as d f ( α p ) + λ =, =,,,. (7) dp Solving + simultaneous equations in () and (7), we can calculate the optimal set of the transmit power {p }. As mentioned above, the function in (4) is a function determined by a specific modulation scheme. For differential phase-shift eying (DPS) scheme, for example, the function may be expressed as an exponential function [] and a closed-form solution of () and (7) may be easily found. For M-ary phase-shift eying (PS) or M-ary quadrature amplitude modulation (QAM) scheme, however, the exact or approximate function may be expressed as a Q-function [] and it may be difficult to find a closed-form solution. In this case, an adaptive method such as the steepest descent algorithm [7] may be employed to find a solution in an iterative manner as follows. Step ) Initialization: Set an iteration number i =, a step size µ() = µ and an arbitrary initial power set {p ()} satisfying (). Step ) Updating the power set: For =,,,, update the transmit power p (i) as p ( i + ) = p µ J ( p, p,, p ) p (8) d = p µ f ( α p ) + λ dp where λ(i) is determined from the power

3 constraint in () and is updated as d λ = f ( p ) α. (9) dp = Step 3) Adjusting the step size in case of negative power: If all components of the calculated power set in Step are non-negative, then go to Step 4 with µ(i+) = µ. Otherwise, adjust the step size µ(i) so that the negative components can be non-negative, and return to Step. Step 4) Repetition or termination: If more adaptations are required for convergence, increase i by one and go to Step. Otherwise, terminate the adaptive procedure. Note that the adaptive solution described above converges to the global optimum solution for the convex function [8], if the number of iterations is sufficiently large and the step size is sufficiently small. The Q-function, which is the exact or approximate function for M-ary PS or M-ary QAM modulation scheme, is a convex function. C. Power Allocation In case that a closed-form solution cannot be found, an adaptive method may be used to find the optimal solution, as described in the previous subsection. However, it generally taes a lot of iterations for an adaptive solution to be close to the optimal one. A simpler approach is to find an approximate closed-form solution using a simple approximation of the rather than the exact expression. Based on this approach, in this subsection, we derive a closed-form solution for M-ary square (M = 4 m, m =,, ) QAM schemes. The derivation procedure may easily be applied to other modulation schemes with modifications for approximate functions. For an M- ary square QAM, the function in (4) may be approximated using an upper bound as [9], [] a b f ( α p ) aq( bα p ) exp α p () where ( π ) exp( ) x Q x t dt denotes the Q- ( M ) function, a =, and b = 3 ( M ). By M log M substituting the upper bound in () into the function of (7), we can find a closed-form solution of + simultaneous equations in () and (7). Some components of the calculated power set may be negative. In this case, we can apply the uhn-tucer conditions [8], from which the negative components of {p } are set to zero and the remaining components are recalculated until all components become non-negative. Consequently, the solution is given as λ α ( b)( α ) ln ( α ), α exp( bλ / ) p =, α < exp( bλ / ) () 4λ where λ ln. To satisfy the power constraint b ab in (), λ is calculated as P + ( b)( α ) ln ( α ) S λ = () α ( ) S where S { α exp( bλ / ) } =. From (), it can be seen that no transmit power is allocated to the th subcarrier if the channel state is smaller than exp( bλ / ). D. Power Allocation In this subsection, the equal SNR power allocation scheme is derived from an asymptotic solution of (). From (), it can be shown that p approaches λ as goes to infinity for all. Thus, the power allocation and the corresponding SNR may be calculated as P p = λ α =, (3) α α = ( α ) = P γ = λ =, =,,,. (4) Equation (4) indicates that the suboptimal scheme of () behaves lie the equal SNR scheme, when is sufficiently large for all. In other words, the received SNR becomes the same for all subchannels, when all the subchannels are in sufficiently good conditions. This equal SNR scheme allocates transmit power inversely proportional to the channel state, allocating the more transmit power to the more attenuated subchannel. Hence, as compared to the equal power scheme, the equal SNR scheme increases the received SNR of relatively worse subchannels, while decreases that of relatively better subchannels. IV. NUMERICAL RESULTS In this section, the performance of the power allocation schemes described in Section III are evaluated and compared with one another. An M-ary square QAM modulation is assumed to be employed for each subcarrier, and adaptive procedure in Section III-B is used to calculate the allocated power for the optimal scheme with sufficient iterations. Equations () and () are used to determine the allocated power for the suboptimal scheme, and equation (3) is used for the equal SNR scheme. The average SNR is defined to be P / σ. The average for each transmit power allocation scheme is calculated by averaging the in (5) for sufficient number of randomly generated channel states { α =,,, }. The characteristics of power allocation schemes are depicted in Fig., which shows the power calculated from three power allocation schemes for given values of, when the number of subcarriers =, modulation level M = 4, and P =. Note that larger value of corresponds to lower attenuation of the combined channel for the th subcarrier. The optimal power allocation

4 scheme is observed to assign the more transmit power to the less attenuated subchannel, when is smaller than an intermediate value,.6 in Fig.. For greater than.6, on the contrary, the more transmit power is assigned to the more attenuated subchannel. Hence, the optimal power allocation may be viewed as a combination of inverse water-filling and water-filling strategies. The trends of the suboptimal power allocation scheme are shown to be similar to that of the optimal power allocation scheme, except that the power corresponding to some highly attenuated subchannels is forced to zero. As expected, the equal SNR scheme allocates the more transmit power to the more attenuated subchannel. Fig. 3 compares the performance of the transmit power allocation schemes for various modulation level M, when = 5 and N =. As expected, the optimal power allocation scheme provides the best performance for all cases. In the case of QPS modulation (M = 4), for example, the SNR gain of the optimal scheme is about 3. db over the equal power scheme, and. db over the equal SNR scheme at a of -3. It is noticeable that the performance of the optimal and suboptimal schemes is almost indistinguishable at high SNR range, especially for low modulation levels. At low SNR range, however, the curve of the suboptimal scheme significantly deviates from that of the optimal scheme, since the difference between the exact and approximate becomes significant with SNR decreasing. The performance improvement of the optimal and suboptimal schemes over the other schemes is found to be a little larger for lower modulation level. The equal SNR scheme outperforms the equal power scheme at high SNR range, while the trend is reversed at low SNR range. This may be a consequence of two conflicting effects of the equal SNR scheme compared to the equal power scheme. As mentioned in Section III-D, the equal SNR scheme increases the received SNR of relatively worse subchannels, while decreases that of relatively better subchannels. Numerical results in Fig. 3 indicates that the effect of the former on the overall performance is more significant at high SNR range, while that of the latter is more significant at low SNR range. The effects of the number of receiving antennas N on the performance are shown in Fig. 4, when M = 4 and = 4. As shown in Fig. 3, the optimal and suboptimal schemes significantly outperform the equal power scheme, for any value of N. We can observe that the equal SNR scheme also outperforms the equal power scheme, unless the SNR is very low or the number of receiving antennas is small. Furthermore, the performance of the equal SNR scheme is found to approach that of the optimal scheme, as N increases. This implies that increased diversity effects resulting from more receiving antennas provide higher probability of the combined channels being in sufficiently good condition for all subcarriers. Hence, the use of equal SNR scheme may be desirable, when the number of receiving antennas is sufficiently large. Fig. 5 shows the effects of the number of subcarriers on the performance, when M = 4 and N =. Note that the performance of the equal power scheme is irrespective of. The SNR gains of the optimal and equal SNR schemes over the equal power scheme are found to increase with increasing. Note that greater also maes the for the proposed power allocation schemes decline more rapidly with SNR increasing. This phenomenon indicates that an increase in the number of subchannels can provide additional diversity effects through the use of proposed power allocation schemes. V. CONCLUSIONS In this paper, we have developed the optimal transmit power allocation scheme that improves the performance in a wireless multicarrier system with diversity reception. A computationally efficient suboptimal scheme has also been derived for M-ary QAM, and the equal SNR scheme has been shown to be an asymptotic case of the suboptimal scheme. Numerical results have shown that the optimal power allocation scheme provides about 3. db of SNR gain over the equal power scheme at a of -3 for 5 subchannels, receiving antennas, and QPS modulation. The performance improvement of the optimal scheme has been found to increase, as the number of subchannels increases. The suboptimal scheme performs as well as the optimal scheme at high SNR range. It has also been found that the performance of the equal SNR scheme approaches that of the optimal scheme, as the number of receiving antennas increases. REFERENCES [] J. G. Proais, Digital Communications. New Yor: McGraw-Hill, 995. [] A. A. Hutter, J. S. Hammerschmidt, E. Carvalho, and J. M. Cioffi, Receive diversity for mobile OFDM systems, in Proc. IEEE WCNC, Chicago, USA, Sept., pp [3] T. M. Cover and J. A. Thomas, Elements of Information Theory. New Yor: Wiley, 99. [4] J. Jang,. B. Lee, and Y.-H. Lee, Frequency-time domain transmit power adaptation for a multicarrier system in fading channels, in Proc. PI, San Diego, USA, Sept.-Oct., pp. D D3. [5] A. J. Goldsmith and S.-G. Chua, Variable-rate variablepower MQAM for fading channels, IEEE Trans. Commun., vol. 45, pp. 8-3, Oct [6] S. T. Chung and A. J. Goldsmith, Adaptive multicarrier modulation for wireless systems, in Proc. 34th Asilomar Conf. Signals, Syst., Comput.,, vol., pp [7] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 985. [8] D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 973. [9] M.. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques. Englewood Cliffs, NJ: Prentice-Hall, 995. [] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New Yor: Wiley, 965.

5 FIGURES d Transmitter Power Allocation Channel # y, y, y, Receiver ^d Input data Serial-toparallel converter d d p p p modulator # #N y, y, y, y,n y,n y,n ^ d ^ d Paralleltoserial converter Output data Fig.. communication system with receive antenna diversity N = N = 4 - p 3-3 N = 3-4 Equal Power - α Fig.. Power allocation for M = 4, =, and P = Fig. 4. Effects of the number of receiving antennas N on the performance of power allocation schemes, when M = 4 and = 4. - M = 4 M = 6 M = Equal Power -3-4 Equal Power (=3,5,7) (=3) (=3) (=5) (=5) (=7) (=7) Fig. 3. performance comparisons of power allocation schemes for = 5 and N = Fig. 5. Effects of the number of subcarriers on the performance of power allocation schemes, when M = 4 and N =.

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